Regression with Label Permutation in Generalized Linear Model

TL;DR

This paper analyzes label permutation issues in generalized linear models with multivariate responses, proposing theoretical results and two algorithms that effectively recover true labels under various knowledge scenarios, including missing data.

Contribution

It provides a comprehensive theoretical framework for label permutation problems in GLMs, extending to missing data and proposing practical algorithms.

Findings

Both algorithms perform well when label permutation proportion is moderate.

Theoretical results remove previous stringent conditions.

Numerical experiments support the effectiveness of the proposed methods.

Abstract

The assumption that response and predictor belong to the same statistical unit may be violated in practice. Unbiased estimation and recovery of true label ordering based on unlabeled data are challenging tasks and have attracted increasing attentions in the recent literature. In this paper, we present a relatively complete analysis of label permutation problem for the generalized linear model with multivariate responses. The theory is established under different scenarios, with knowledge of true parameters, with partial knowledge of underlying label permutation matrix and without any knowledge. Our results remove the stringent conditions required by the current literature and are further extended to the missing observation setting which has never been considered in the field of label permutation problem. On computational side, we propose two methods, "maximum likelihood estimation" algorithm and "two-step estimation" algorithm, to accommodate for different settings. When the proportion of permuted labels is moderate, both methods work effectively. Multiple numerical experiments are provided and corroborate our theoretical findings.